Structural abstract interpretation, A formal study using Coq

interpreters are tools to compute approximations for behaviors of a program. These approximations can then be used for optimisation or for error detection. In this paper, we show how to describe an abstract interpreter using the type-theory based theorem prover Coq, using inductive types for syntax and structural recursive programming for the abstract interpreter's kernel. The abstract interpreter can then be proved correct with respect to a Hoare logic for the programming language

Collection analysis for Horn clause programs

We consider approximating data structures with collections of the items that they contain. For examples, lists, binary trees, tuples, etc, can be approximated by sets or multisets of the items within them. Such approximations can be used to provide partial correctness properties of logic programs. For example, one might wish to specify than whenever the atom $sort(t,s)$ is proved then the two lists $t$ and $s$ contain the same multiset of items (that is, $s$ is a permutation of $t$). If sorting removes duplicates, then one would like to infer that the sets of items underlying $t$ and $s$ are the same. Such results could be useful to have if they can be determined statically and automatically. We present a scheme by which such collection analysis can be structured and automated. Central to this scheme is the use of linear logic as a omputational logic underlying the logic of Horn clauses

Lifted Relax, Compensate and then Recover: From Approximate to Exact Lifted Probabilistic Inference

We propose an approach to lifted approximate inference for first-order probabilistic models, such as Markov logic networks. It is based on performing exact lifted inference in a simplified first-order model, which is found by relaxing first-order constraints, and then compensating for the relaxation. These simplified models can be incrementally improved by carefully recovering constraints that have been relaxed, also at the first-order level. This leads to a spectrum of approximations, with lifted belief propagation on one end, and exact lifted inference on the other. We discuss how relaxation, compensation, and recovery can be performed, all at the firstorder level, and show empirically that our approach substantially improves on the approximations of both propositional solvers and lifted belief propagation.Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002

Duality in Logic Programming

Various approximations of classic negation have been proposed for logic programming. But the semantics for those approximations are not entirely clear. In this paper a proof-theoretic operator, we call it failure operator, denoted as FP, is associated with each logic program to characterize the meaning of various negations in logic programming. It is shown that the failure operator FP is a dual of the TP, immediate consequence operator developed by Van Emden and Kowalski and is downward continuous. It has the desirable properties entirely analogous to what TP has such as continuity, having a unique least fixpoint and a unique greatest fixpoint. It provides natural proof theories for various version negations in logic programming. We prove that set complementation provides the isomorphism between the fixpoints of FP and those of TP, which illustrates the duality of FP and TP. The existing treatment of negation in logic programming can be given in a simple and elegant fixpoint characterization